Weak and strong convergence of an iterative algorithm for Lipschitz pseudo-contractive maps in Hilbert spaces

Abstract

In this paper, let K be a closed convex subset of a real Hilbert space H and T: K → K a Lipschitz pseudo-contractive map such that F(T) 6= / 0. Let {αn}, {βn} and {γn} be real sequences in (0,1). For x1∈ K, let {xn} be generated iteratively by

xn+1= Pk[(1−αn−γn)xn+γnTyn],

yn= (1−βn)xn+βnTxn,n ≥ 1.

Under some mild conditions on parameters {αn}, {βn}, {γn}, we prove that our new iterative algorithm converges strongly to a fixed point of T. No compactness assumption is imposed on T and no further requirement is imposed on F(T).